Variations on a theorem by Alan Camina on conjugacy class sizes

Abstract

Let G be a finite group. We extend Alan Camina's theorem on conjugacy class sizes which asserts that if the conjugacy class sizes of G are exactly { 1 , p a , q b , p a q b } for two primes p and q, then G is nilpotent. If we assume that G is solvable, we show that when the set of conjugacy class sizes of G is { 1 , m , n , m n } with m and n arbitrary positive integers such that ( m , n ) = 1 , then G is nilpotent and m = p a and n = q b for two primes p and q.